LMFDB - Embedded newform 8046.2.a.q.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.39851$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.39851 q^{5} +2.84704 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.39851 q^{5} +2.84704 q^{7} -1.00000 q^{8} -2.39851 q^{10} -5.71165 q^{11} -5.33016 q^{13} -2.84704 q^{14} +1.00000 q^{16} +5.68131 q^{17} -1.71918 q^{19} +2.39851 q^{20} +5.71165 q^{22} -6.10300 q^{23} +0.752832 q^{25} +5.33016 q^{26} +2.84704 q^{28} +2.56079 q^{29} +4.56683 q^{31} -1.00000 q^{32} -5.68131 q^{34} +6.82865 q^{35} +8.20682 q^{37} +1.71918 q^{38} -2.39851 q^{40} -0.830268 q^{41} +3.47655 q^{43} -5.71165 q^{44} +6.10300 q^{46} +3.29150 q^{47} +1.10565 q^{49} -0.752832 q^{50} -5.33016 q^{52} -10.9710 q^{53} -13.6994 q^{55} -2.84704 q^{56} -2.56079 q^{58} -12.2558 q^{59} -4.73678 q^{61} -4.56683 q^{62} +1.00000 q^{64} -12.7844 q^{65} +1.17659 q^{67} +5.68131 q^{68} -6.82865 q^{70} +10.5859 q^{71} +13.4736 q^{73} -8.20682 q^{74} -1.71918 q^{76} -16.2613 q^{77} +13.6549 q^{79} +2.39851 q^{80} +0.830268 q^{82} +9.57663 q^{83} +13.6267 q^{85} -3.47655 q^{86} +5.71165 q^{88} +4.62984 q^{89} -15.1752 q^{91} -6.10300 q^{92} -3.29150 q^{94} -4.12345 q^{95} +8.89340 q^{97} -1.10565 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.39851	1.07264	0.536322	−	0.844013i	$-0.319813\pi$
$5$	2.39851	1.07264	0.536322	+	0.844013i	$0.319813\pi$
$6$	0	0
$7$	2.84704	1.07608	0.538040	−	0.842919i	$-0.319165\pi$
$7$	2.84704	1.07608	0.538040	+	0.842919i	$0.319165\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.39851	−0.758474
$11$	−5.71165	−1.72213	−0.861063	−	0.508498i	$-0.830201\pi$
$11$	−5.71165	−1.72213	−0.861063	+	0.508498i	$0.830201\pi$
$12$	0	0
$13$	−5.33016	−1.47832	−0.739160	−	0.673530i	$-0.764777\pi$
$13$	−5.33016	−1.47832	−0.739160	+	0.673530i	$0.764777\pi$
$14$	−2.84704	−0.760904
$15$	0	0
$16$	1.00000	0.250000
$17$	5.68131	1.37792	0.688960	−	0.724799i	$-0.258067\pi$
$17$	5.68131	1.37792	0.688960	+	0.724799i	$0.258067\pi$
$18$	0	0
$19$	−1.71918	−0.394406	−0.197203	−	0.980363i	$-0.563186\pi$
$19$	−1.71918	−0.394406	−0.197203	+	0.980363i	$0.563186\pi$
$20$	2.39851	0.536322
$21$	0	0
$22$	5.71165	1.21773
$23$	−6.10300	−1.27256	−0.636282	−	0.771457i	$-0.719528\pi$
$23$	−6.10300	−1.27256	−0.636282	+	0.771457i	$0.719528\pi$
$24$	0	0
$25$	0.752832	0.150566
$26$	5.33016	1.04533
$27$	0	0
$28$	2.84704	0.538040
$29$	2.56079	0.475526	0.237763	−	0.971323i	$-0.423586\pi$
$29$	2.56079	0.475526	0.237763	+	0.971323i	$0.423586\pi$
$30$	0	0
$31$	4.56683	0.820226	0.410113	−	0.912035i	$-0.365489\pi$
$31$	4.56683	0.820226	0.410113	+	0.912035i	$0.365489\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.68131	−0.974337
$35$	6.82865	1.15425
$36$	0	0
$37$	8.20682	1.34919	0.674596	−	0.738187i	$-0.264318\pi$
$37$	8.20682	1.34919	0.674596	+	0.738187i	$0.264318\pi$
$38$	1.71918	0.278887
$39$	0	0
$40$	−2.39851	−0.379237
$41$	−0.830268	−0.129666	−0.0648330	−	0.997896i	$-0.520651\pi$
$41$	−0.830268	−0.129666	−0.0648330	+	0.997896i	$0.520651\pi$
$42$	0	0
$43$	3.47655	0.530169	0.265085	−	0.964225i	$-0.414600\pi$
$43$	3.47655	0.530169	0.265085	+	0.964225i	$0.414600\pi$
$44$	−5.71165	−0.861063
$45$	0	0
$46$	6.10300	0.899838
$47$	3.29150	0.480115	0.240057	−	0.970759i	$-0.422834\pi$
$47$	3.29150	0.480115	0.240057	+	0.970759i	$0.422834\pi$
$48$	0	0
$49$	1.10565	0.157950
$50$	−0.752832	−0.106467
$51$	0	0
$52$	−5.33016	−0.739160
$53$	−10.9710	−1.50698	−0.753491	−	0.657458i	$-0.771632\pi$
$53$	−10.9710	−1.50698	−0.753491	+	0.657458i	$0.771632\pi$
$54$	0	0
$55$	−13.6994	−1.84723
$56$	−2.84704	−0.380452
$57$	0	0
$58$	−2.56079	−0.336248
$59$	−12.2558	−1.59558	−0.797788	−	0.602939i	$-0.793996\pi$
$59$	−12.2558	−1.59558	−0.797788	+	0.602939i	$0.793996\pi$
$60$	0	0
$61$	−4.73678	−0.606482	−0.303241	−	0.952914i	$-0.598069\pi$
$61$	−4.73678	−0.606482	−0.303241	+	0.952914i	$0.598069\pi$
$62$	−4.56683	−0.579988
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.7844	−1.58571
$66$	0	0
$67$	1.17659	0.143744	0.0718719	−	0.997414i	$-0.477103\pi$
$67$	1.17659	0.143744	0.0718719	+	0.997414i	$0.477103\pi$
$68$	5.68131	0.688960
$69$	0	0
$70$	−6.82865	−0.816180
$71$	10.5859	1.25631	0.628157	−	0.778087i	$-0.283810\pi$
$71$	10.5859	1.25631	0.628157	+	0.778087i	$0.283810\pi$
$72$	0	0
$73$	13.4736	1.57697	0.788485	−	0.615055i	$-0.210866\pi$
$73$	13.4736	1.57697	0.788485	+	0.615055i	$0.210866\pi$
$74$	−8.20682	−0.954023
$75$	0	0
$76$	−1.71918	−0.197203
$77$	−16.2613	−1.85315
$78$	0	0
$79$	13.6549	1.53630	0.768150	−	0.640270i	$-0.221178\pi$
$79$	13.6549	1.53630	0.768150	+	0.640270i	$0.221178\pi$
$80$	2.39851	0.268161
$81$	0	0
$82$	0.830268	0.0916878
$83$	9.57663	1.05117	0.525586	−	0.850741i	$-0.323846\pi$
$83$	9.57663	1.05117	0.525586	+	0.850741i	$0.323846\pi$
$84$	0	0
$85$	13.6267	1.47802
$86$	−3.47655	−0.374886
$87$	0	0
$88$	5.71165	0.608863
$89$	4.62984	0.490762	0.245381	−	0.969427i	$-0.421087\pi$
$89$	4.62984	0.490762	0.245381	+	0.969427i	$0.421087\pi$
$90$	0	0
$91$	−15.1752	−1.59079
$92$	−6.10300	−0.636282
$93$	0	0
$94$	−3.29150	−0.339492
$95$	−4.12345	−0.423057
$96$	0	0
$97$	8.89340	0.902988	0.451494	−	0.892274i	$-0.350891\pi$
$97$	8.89340	0.902988	0.451494	+	0.892274i	$0.350891\pi$
$98$	−1.10565	−0.111688
$99$	0	0
$100$	0.752832	0.0752832
$101$	19.2296	1.91342	0.956710	−	0.291043i	$-0.0940024\pi$
$101$	19.2296	1.91342	0.956710	+	0.291043i	$0.0940024\pi$
$102$	0	0
$103$	7.07500	0.697120	0.348560	−	0.937286i	$-0.386671\pi$
$103$	7.07500	0.697120	0.348560	+	0.937286i	$0.386671\pi$
$104$	5.33016	0.522665
$105$	0	0
$106$	10.9710	1.06560
$107$	−4.20399	−0.406415	−0.203208	−	0.979136i	$-0.565137\pi$
$107$	−4.20399	−0.406415	−0.203208	+	0.979136i	$0.565137\pi$
$108$	0	0
$109$	−5.10836	−0.489292	−0.244646	−	0.969612i	$-0.578672\pi$
$109$	−5.10836	−0.489292	−0.244646	+	0.969612i	$0.578672\pi$
$110$	13.6994	1.30619
$111$	0	0
$112$	2.84704	0.269020
$113$	15.9176	1.49741	0.748703	−	0.662905i	$-0.230677\pi$
$113$	15.9176	1.49741	0.748703	+	0.662905i	$0.230677\pi$
$114$	0	0
$115$	−14.6381	−1.36501
$116$	2.56079	0.237763
$117$	0	0
$118$	12.2558	1.12824
$119$	16.1749	1.48275
$120$	0	0
$121$	21.6229	1.96572
$122$	4.73678	0.428848
$123$	0	0
$124$	4.56683	0.410113
$125$	−10.1869	−0.911140
$126$	0	0
$127$	5.27783	0.468332	0.234166	−	0.972197i	$-0.424764\pi$
$127$	5.27783	0.468332	0.234166	+	0.972197i	$0.424764\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.7844	1.12127
$131$	−14.5858	−1.27437	−0.637184	−	0.770711i	$-0.719901\pi$
$131$	−14.5858	−1.27437	−0.637184	+	0.770711i	$0.719901\pi$
$132$	0	0
$133$	−4.89457	−0.424413
$134$	−1.17659	−0.101642
$135$	0	0
$136$	−5.68131	−0.487169
$137$	13.2006	1.12780	0.563902	−	0.825842i	$-0.309300\pi$
$137$	13.2006	1.12780	0.563902	+	0.825842i	$0.309300\pi$
$138$	0	0
$139$	16.2736	1.38031	0.690155	−	0.723662i	$-0.257543\pi$
$139$	16.2736	1.38031	0.690155	+	0.723662i	$0.257543\pi$
$140$	6.82865	0.577126
$141$	0	0
$142$	−10.5859	−0.888348
$143$	30.4440	2.54585
$144$	0	0
$145$	6.14206	0.510070
$146$	−13.4736	−1.11509
$147$	0	0
$148$	8.20682	0.674596
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	11.5048	0.936248	0.468124	−	0.883663i	$-0.344930\pi$
$151$	11.5048	0.936248	0.468124	+	0.883663i	$0.344930\pi$
$152$	1.71918	0.139444
$153$	0	0
$154$	16.2613	1.31037
$155$	10.9536	0.879811
$156$	0	0
$157$	21.1964	1.69166	0.845830	−	0.533453i	$-0.179106\pi$
$157$	21.1964	1.69166	0.845830	+	0.533453i	$0.179106\pi$
$158$	−13.6549	−1.08633
$159$	0	0
$160$	−2.39851	−0.189619
$161$	−17.3755	−1.36938
$162$	0	0
$163$	8.69588	0.681114	0.340557	−	0.940224i	$-0.389384\pi$
$163$	8.69588	0.681114	0.340557	+	0.940224i	$0.389384\pi$
$164$	−0.830268	−0.0648330
$165$	0	0
$166$	−9.57663	−0.743290
$167$	−14.1225	−1.09283	−0.546416	−	0.837514i	$-0.684008\pi$
$167$	−14.1225	−1.09283	−0.546416	+	0.837514i	$0.684008\pi$
$168$	0	0
$169$	15.4106	1.18543
$170$	−13.6267	−1.04512
$171$	0	0
$172$	3.47655	0.265085
$173$	5.32450	0.404814	0.202407	−	0.979301i	$-0.435124\pi$
$173$	5.32450	0.404814	0.202407	+	0.979301i	$0.435124\pi$
$174$	0	0
$175$	2.14335	0.162022
$176$	−5.71165	−0.430531
$177$	0	0
$178$	−4.62984	−0.347021
$179$	−0.0832261	−0.00622061	−0.00311031	−	0.999995i	$-0.500990\pi$
$179$	−0.0832261	−0.00622061	−0.00311031	+	0.999995i	$0.500990\pi$
$180$	0	0
$181$	10.0279	0.745366	0.372683	−	0.927959i	$-0.378438\pi$
$181$	10.0279	0.745366	0.372683	+	0.927959i	$0.378438\pi$
$182$	15.1752	1.12486
$183$	0	0
$184$	6.10300	0.449919
$185$	19.6841	1.44720
$186$	0	0
$187$	−32.4496	−2.37295
$188$	3.29150	0.240057
$189$	0	0
$190$	4.12345	0.299147
$191$	13.2145	0.956166	0.478083	−	0.878315i	$-0.341332\pi$
$191$	13.2145	0.956166	0.478083	+	0.878315i	$0.341332\pi$
$192$	0	0
$193$	10.5122	0.756682	0.378341	−	0.925666i	$-0.376495\pi$
$193$	10.5122	0.756682	0.378341	+	0.925666i	$0.376495\pi$
$194$	−8.89340	−0.638509
$195$	0	0
$196$	1.10565	0.0789750
$197$	−18.6356	−1.32773	−0.663864	−	0.747853i	$-0.731085\pi$
$197$	−18.6356	−1.32773	−0.663864	+	0.747853i	$0.731085\pi$
$198$	0	0
$199$	−21.2346	−1.50528	−0.752640	−	0.658433i	$-0.771220\pi$
$199$	−21.2346	−1.50528	−0.752640	+	0.658433i	$0.771220\pi$
$200$	−0.752832	−0.0532333
$201$	0	0
$202$	−19.2296	−1.35299
$203$	7.29067	0.511704
$204$	0	0
$205$	−1.99140	−0.139086
$206$	−7.07500	−0.492938
$207$	0	0
$208$	−5.33016	−0.369580
$209$	9.81932	0.679217
$210$	0	0
$211$	−26.0464	−1.79311	−0.896555	−	0.442932i	$-0.853938\pi$
$211$	−26.0464	−1.79311	−0.896555	+	0.442932i	$0.853938\pi$
$212$	−10.9710	−0.753491
$213$	0	0
$214$	4.20399	0.287379
$215$	8.33853	0.568683
$216$	0	0
$217$	13.0019	0.882630
$218$	5.10836	0.345982
$219$	0	0
$220$	−13.6994	−0.923615
$221$	−30.2823	−2.03701
$222$	0	0
$223$	5.34883	0.358184	0.179092	−	0.983832i	$-0.442684\pi$
$223$	5.34883	0.358184	0.179092	+	0.983832i	$0.442684\pi$
$224$	−2.84704	−0.190226
$225$	0	0
$226$	−15.9176	−1.05883
$227$	12.6760	0.841339	0.420669	−	0.907214i	$-0.361795\pi$
$227$	12.6760	0.841339	0.420669	+	0.907214i	$0.361795\pi$
$228$	0	0
$229$	15.5735	1.02913	0.514563	−	0.857453i	$-0.327954\pi$
$229$	15.5735	1.02913	0.514563	+	0.857453i	$0.327954\pi$
$230$	14.6381	0.965207
$231$	0	0
$232$	−2.56079	−0.168124
$233$	−17.3342	−1.13560	−0.567799	−	0.823167i	$-0.692205\pi$
$233$	−17.3342	−1.13560	−0.567799	+	0.823167i	$0.692205\pi$
$234$	0	0
$235$	7.89469	0.514993
$236$	−12.2558	−0.797788
$237$	0	0
$238$	−16.1749	−1.04847
$239$	11.9581	0.773506	0.386753	−	0.922183i	$-0.373597\pi$
$239$	11.9581	0.773506	0.386753	+	0.922183i	$0.373597\pi$
$240$	0	0
$241$	8.21050	0.528885	0.264442	−	0.964401i	$-0.414812\pi$
$241$	8.21050	0.528885	0.264442	+	0.964401i	$0.414812\pi$
$242$	−21.6229	−1.38997
$243$	0	0
$244$	−4.73678	−0.303241
$245$	2.65191	0.169424
$246$	0	0
$247$	9.16347	0.583058
$248$	−4.56683	−0.289994
$249$	0	0
$250$	10.1869	0.644274
$251$	−3.58233	−0.226115	−0.113057	−	0.993588i	$-0.536064\pi$
$251$	−3.58233	−0.226115	−0.113057	+	0.993588i	$0.536064\pi$
$252$	0	0
$253$	34.8582	2.19151
$254$	−5.27783	−0.331161
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.32245	−0.394384	−0.197192	−	0.980365i	$-0.563182\pi$
$257$	−6.32245	−0.394384	−0.197192	+	0.980365i	$0.563182\pi$
$258$	0	0
$259$	23.3652	1.45184
$260$	−12.7844	−0.792856
$261$	0	0
$262$	14.5858	0.901115
$263$	−11.0838	−0.683454	−0.341727	−	0.939799i	$-0.611012\pi$
$263$	−11.0838	−0.683454	−0.341727	+	0.939799i	$0.611012\pi$
$264$	0	0
$265$	−26.3140	−1.61646
$266$	4.89457	0.300105
$267$	0	0
$268$	1.17659	0.0718719
$269$	6.53730	0.398586	0.199293	−	0.979940i	$-0.436135\pi$
$269$	6.53730	0.398586	0.199293	+	0.979940i	$0.436135\pi$
$270$	0	0
$271$	−24.3298	−1.47793	−0.738965	−	0.673744i	$-0.764685\pi$
$271$	−24.3298	−1.47793	−0.738965	+	0.673744i	$0.764685\pi$
$272$	5.68131	0.344480
$273$	0	0
$274$	−13.2006	−0.797478
$275$	−4.29991	−0.259294
$276$	0	0
$277$	11.1643	0.670797	0.335399	−	0.942076i	$-0.391129\pi$
$277$	11.1643	0.670797	0.335399	+	0.942076i	$0.391129\pi$
$278$	−16.2736	−0.976026
$279$	0	0
$280$	−6.82865	−0.408090
$281$	11.6434	0.694587	0.347294	−	0.937756i	$-0.387101\pi$
$281$	11.6434	0.694587	0.347294	+	0.937756i	$0.387101\pi$
$282$	0	0
$283$	−14.9323	−0.887636	−0.443818	−	0.896117i	$-0.646376\pi$
$283$	−14.9323	−0.887636	−0.443818	+	0.896117i	$0.646376\pi$
$284$	10.5859	0.628157
$285$	0	0
$286$	−30.4440	−1.80019
$287$	−2.36381	−0.139531
$288$	0	0
$289$	15.2773	0.898666
$290$	−6.14206	−0.360674
$291$	0	0
$292$	13.4736	0.788485
$293$	−1.52528	−0.0891080	−0.0445540	−	0.999007i	$-0.514187\pi$
$293$	−1.52528	−0.0891080	−0.0445540	+	0.999007i	$0.514187\pi$
$294$	0	0
$295$	−29.3957	−1.71149
$296$	−8.20682	−0.477011
$297$	0	0
$298$	1.00000	0.0579284
$299$	32.5299	1.88125
$300$	0	0
$301$	9.89789	0.570505
$302$	−11.5048	−0.662027
$303$	0	0
$304$	−1.71918	−0.0986015
$305$	−11.3612	−0.650540
$306$	0	0
$307$	−3.02894	−0.172870	−0.0864352	−	0.996257i	$-0.527548\pi$
$307$	−3.02894	−0.172870	−0.0864352	+	0.996257i	$0.527548\pi$
$308$	−16.2613	−0.926573
$309$	0	0
$310$	−10.9536	−0.622121
$311$	11.2549	0.638207	0.319103	−	0.947720i	$-0.396618\pi$
$311$	11.2549	0.638207	0.319103	+	0.947720i	$0.396618\pi$
$312$	0	0
$313$	29.8744	1.68860	0.844301	−	0.535869i	$-0.180016\pi$
$313$	29.8744	1.68860	0.844301	+	0.535869i	$0.180016\pi$
$314$	−21.1964	−1.19618
$315$	0	0
$316$	13.6549	0.768150
$317$	25.6497	1.44063	0.720316	−	0.693646i	$-0.243997\pi$
$317$	25.6497	1.44063	0.720316	+	0.693646i	$0.243997\pi$
$318$	0	0
$319$	−14.6263	−0.818916
$320$	2.39851	0.134081
$321$	0	0
$322$	17.3755	0.968299
$323$	−9.76718	−0.543460
$324$	0	0
$325$	−4.01271	−0.222585
$326$	−8.69588	−0.481621
$327$	0	0
$328$	0.830268	0.0458439
$329$	9.37104	0.516642
$330$	0	0
$331$	−14.9538	−0.821936	−0.410968	−	0.911650i	$-0.634809\pi$
$331$	−14.9538	−0.821936	−0.410968	+	0.911650i	$0.634809\pi$
$332$	9.57663	0.525586
$333$	0	0
$334$	14.1225	0.772750
$335$	2.82207	0.154186
$336$	0	0
$337$	22.2886	1.21414	0.607068	−	0.794650i	$-0.292346\pi$
$337$	22.2886	1.21414	0.607068	+	0.794650i	$0.292346\pi$
$338$	−15.4106	−0.838224
$339$	0	0
$340$	13.6267	0.739010
$341$	−26.0841	−1.41253
$342$	0	0
$343$	−16.7815	−0.906114
$344$	−3.47655	−0.187443
$345$	0	0
$346$	−5.32450	−0.286247
$347$	−3.31106	−0.177747	−0.0888734	−	0.996043i	$-0.528327\pi$
$347$	−3.31106	−0.177747	−0.0888734	+	0.996043i	$0.528327\pi$
$348$	0	0
$349$	19.7113	1.05512	0.527562	−	0.849517i	$-0.323106\pi$
$349$	19.7113	1.05512	0.527562	+	0.849517i	$0.323106\pi$
$350$	−2.14335	−0.114567
$351$	0	0
$352$	5.71165	0.304432
$353$	−19.2372	−1.02389	−0.511946	−	0.859018i	$-0.671075\pi$
$353$	−19.2372	−1.02389	−0.511946	+	0.859018i	$0.671075\pi$
$354$	0	0
$355$	25.3903	1.34758
$356$	4.62984	0.245381
$357$	0	0
$358$	0.0832261	0.00439864
$359$	−4.85892	−0.256444	−0.128222	−	0.991745i	$-0.540927\pi$
$359$	−4.85892	−0.256444	−0.128222	+	0.991745i	$0.540927\pi$
$360$	0	0
$361$	−16.0444	−0.844444
$362$	−10.0279	−0.527053
$363$	0	0
$364$	−15.1752	−0.795396
$365$	32.3166	1.69153
$366$	0	0
$367$	−1.25818	−0.0656763	−0.0328381	−	0.999461i	$-0.510455\pi$
$367$	−1.25818	−0.0656763	−0.0328381	+	0.999461i	$0.510455\pi$
$368$	−6.10300	−0.318141
$369$	0	0
$370$	−19.6841	−1.02333
$371$	−31.2349	−1.62163
$372$	0	0
$373$	−21.6434	−1.12065	−0.560325	−	0.828273i	$-0.689324\pi$
$373$	−21.6434	−1.12065	−0.560325	+	0.828273i	$0.689324\pi$
$374$	32.4496	1.67793
$375$	0	0
$376$	−3.29150	−0.169746
$377$	−13.6494	−0.702979
$378$	0	0
$379$	24.9958	1.28395	0.641975	−	0.766726i	$-0.278115\pi$
$379$	24.9958	1.28395	0.641975	+	0.766726i	$0.278115\pi$
$380$	−4.12345	−0.211529
$381$	0	0
$382$	−13.2145	−0.676111
$383$	−24.0367	−1.22822	−0.614109	−	0.789221i	$-0.710484\pi$
$383$	−24.0367	−1.22822	−0.614109	+	0.789221i	$0.710484\pi$
$384$	0	0
$385$	−39.0028	−1.98777
$386$	−10.5122	−0.535055
$387$	0	0
$388$	8.89340	0.451494
$389$	−7.31780	−0.371027	−0.185514	−	0.982642i	$-0.559395\pi$
$389$	−7.31780	−0.371027	−0.185514	+	0.982642i	$0.559395\pi$
$390$	0	0
$391$	−34.6731	−1.75349
$392$	−1.10565	−0.0558438
$393$	0	0
$394$	18.6356	0.938846
$395$	32.7514	1.64790
$396$	0	0
$397$	−22.6229	−1.13541	−0.567705	−	0.823232i	$-0.692169\pi$
$397$	−22.6229	−1.13541	−0.567705	+	0.823232i	$0.692169\pi$
$398$	21.2346	1.06439
$399$	0	0
$400$	0.752832	0.0376416
$401$	−38.9060	−1.94287	−0.971436	−	0.237303i	$-0.923737\pi$
$401$	−38.9060	−1.94287	−0.971436	+	0.237303i	$0.923737\pi$
$402$	0	0
$403$	−24.3419	−1.21256
$404$	19.2296	0.956710
$405$	0	0
$406$	−7.29067	−0.361830
$407$	−46.8744	−2.32348
$408$	0	0
$409$	−4.29130	−0.212191	−0.106096	−	0.994356i	$-0.533835\pi$
$409$	−4.29130	−0.212191	−0.106096	+	0.994356i	$0.533835\pi$
$410$	1.99140	0.0983484
$411$	0	0
$412$	7.07500	0.348560
$413$	−34.8929	−1.71697
$414$	0	0
$415$	22.9696	1.12753
$416$	5.33016	0.261332
$417$	0	0
$418$	−9.81932	−0.480279
$419$	−28.9077	−1.41224	−0.706118	−	0.708094i	$-0.749555\pi$
$419$	−28.9077	−1.41224	−0.706118	+	0.708094i	$0.749555\pi$
$420$	0	0
$421$	9.14857	0.445874	0.222937	−	0.974833i	$-0.428436\pi$
$421$	9.14857	0.445874	0.222937	+	0.974833i	$0.428436\pi$
$422$	26.0464	1.26792
$423$	0	0
$424$	10.9710	0.532799
$425$	4.27708	0.207469
$426$	0	0
$427$	−13.4858	−0.652624
$428$	−4.20399	−0.203208
$429$	0	0
$430$	−8.33853	−0.402120
$431$	3.96808	0.191136	0.0955678	−	0.995423i	$-0.469533\pi$
$431$	3.96808	0.191136	0.0955678	+	0.995423i	$0.469533\pi$
$432$	0	0
$433$	−7.30067	−0.350848	−0.175424	−	0.984493i	$-0.556130\pi$
$433$	−7.30067	−0.350848	−0.175424	+	0.984493i	$0.556130\pi$
$434$	−13.0019	−0.624114
$435$	0	0
$436$	−5.10836	−0.244646
$437$	10.4921	0.501906
$438$	0	0
$439$	16.0443	0.765751	0.382875	−	0.923800i	$-0.374934\pi$
$439$	16.0443	0.765751	0.382875	+	0.923800i	$0.374934\pi$
$440$	13.6994	0.653094
$441$	0	0
$442$	30.2823	1.44038
$443$	−1.54926	−0.0736076	−0.0368038	−	0.999323i	$-0.511718\pi$
$443$	−1.54926	−0.0736076	−0.0368038	+	0.999323i	$0.511718\pi$
$444$	0	0
$445$	11.1047	0.526413
$446$	−5.34883	−0.253274
$447$	0	0
$448$	2.84704	0.134510
$449$	1.09546	0.0516978	0.0258489	−	0.999666i	$-0.491771\pi$
$449$	1.09546	0.0516978	0.0258489	+	0.999666i	$0.491771\pi$
$450$	0	0
$451$	4.74220	0.223301
$452$	15.9176	0.748703
$453$	0	0
$454$	−12.6760	−0.594916
$455$	−36.3978	−1.70635
$456$	0	0
$457$	6.70058	0.313440	0.156720	−	0.987643i	$-0.449908\pi$
$457$	6.70058	0.313440	0.156720	+	0.987643i	$0.449908\pi$
$458$	−15.5735	−0.727701
$459$	0	0
$460$	−14.6381	−0.682504
$461$	−32.2767	−1.50328	−0.751638	−	0.659576i	$-0.770736\pi$
$461$	−32.2767	−1.50328	−0.751638	+	0.659576i	$0.770736\pi$
$462$	0	0
$463$	21.3006	0.989922	0.494961	−	0.868915i	$-0.335182\pi$
$463$	21.3006	0.989922	0.494961	+	0.868915i	$0.335182\pi$
$464$	2.56079	0.118881
$465$	0	0
$466$	17.3342	0.802989
$467$	3.88798	0.179914	0.0899572	−	0.995946i	$-0.471327\pi$
$467$	3.88798	0.179914	0.0899572	+	0.995946i	$0.471327\pi$
$468$	0	0
$469$	3.34981	0.154680
$470$	−7.89469	−0.364155
$471$	0	0
$472$	12.2558	0.564121
$473$	−19.8568	−0.913018
$474$	0	0
$475$	−1.29425	−0.0593843
$476$	16.1749	0.741377
$477$	0	0
$478$	−11.9581	−0.546951
$479$	−4.05024	−0.185060	−0.0925302	−	0.995710i	$-0.529495\pi$
$479$	−4.05024	−0.185060	−0.0925302	+	0.995710i	$0.529495\pi$
$480$	0	0
$481$	−43.7436	−1.99454
$482$	−8.21050	−0.373978
$483$	0	0
$484$	21.6229	0.982859
$485$	21.3309	0.968586
$486$	0	0
$487$	5.04846	0.228768	0.114384	−	0.993437i	$-0.463511\pi$
$487$	5.04846	0.228768	0.114384	+	0.993437i	$0.463511\pi$
$488$	4.73678	0.214424
$489$	0	0
$490$	−2.65191	−0.119801
$491$	−22.2298	−1.00322	−0.501609	−	0.865094i	$-0.667258\pi$
$491$	−22.2298	−1.00322	−0.501609	+	0.865094i	$0.667258\pi$
$492$	0	0
$493$	14.5486	0.655237
$494$	−9.16347	−0.412284
$495$	0	0
$496$	4.56683	0.205057
$497$	30.1385	1.35189
$498$	0	0
$499$	29.4416	1.31799	0.658993	−	0.752149i	$-0.270983\pi$
$499$	29.4416	1.31799	0.658993	+	0.752149i	$0.270983\pi$
$500$	−10.1869	−0.455570
$501$	0	0
$502$	3.58233	0.159887
$503$	27.8738	1.24283	0.621415	−	0.783482i	$-0.286558\pi$
$503$	27.8738	1.24283	0.621415	+	0.783482i	$0.286558\pi$
$504$	0	0
$505$	46.1224	2.05242
$506$	−34.8582	−1.54963
$507$	0	0
$508$	5.27783	0.234166
$509$	−21.3373	−0.945758	−0.472879	−	0.881127i	$-0.656785\pi$
$509$	−21.3373	−0.945758	−0.472879	+	0.881127i	$0.656785\pi$
$510$	0	0
$511$	38.3600	1.69695
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.32245	0.278871
$515$	16.9694	0.747762
$516$	0	0
$517$	−18.7999	−0.826818
$518$	−23.3652	−1.02661
$519$	0	0
$520$	12.7844	0.560634
$521$	37.8833	1.65970	0.829848	−	0.557989i	$-0.188427\pi$
$521$	37.8833	1.65970	0.829848	+	0.557989i	$0.188427\pi$
$522$	0	0
$523$	33.9588	1.48492	0.742458	−	0.669892i	$-0.233660\pi$
$523$	33.9588	1.48492	0.742458	+	0.669892i	$0.233660\pi$
$524$	−14.5858	−0.637184
$525$	0	0
$526$	11.0838	0.483275
$527$	25.9456	1.13021
$528$	0	0
$529$	14.2466	0.619417
$530$	26.3140	1.14301
$531$	0	0
$532$	−4.89457	−0.212206
$533$	4.42546	0.191688
$534$	0	0
$535$	−10.0833	−0.435939
$536$	−1.17659	−0.0508211
$537$	0	0
$538$	−6.53730	−0.281843
$539$	−6.31508	−0.272010
$540$	0	0
$541$	22.4265	0.964190	0.482095	−	0.876119i	$-0.339876\pi$
$541$	22.4265	0.964190	0.482095	+	0.876119i	$0.339876\pi$
$542$	24.3298	1.04505
$543$	0	0
$544$	−5.68131	−0.243584
$545$	−12.2524	−0.524837
$546$	0	0
$547$	−18.4757	−0.789964	−0.394982	−	0.918689i	$-0.629249\pi$
$547$	−18.4757	−0.789964	−0.394982	+	0.918689i	$0.629249\pi$
$548$	13.2006	0.563902
$549$	0	0
$550$	4.29991	0.183349
$551$	−4.40244	−0.187550
$552$	0	0
$553$	38.8761	1.65318
$554$	−11.1643	−0.474325
$555$	0	0
$556$	16.2736	0.690155
$557$	−18.1474	−0.768930	−0.384465	−	0.923140i	$-0.625614\pi$
$557$	−18.1474	−0.768930	−0.384465	+	0.923140i	$0.625614\pi$
$558$	0	0
$559$	−18.5306	−0.783760
$560$	6.82865	0.288563
$561$	0	0
$562$	−11.6434	−0.491147
$563$	39.9625	1.68422	0.842109	−	0.539308i	$-0.181314\pi$
$563$	39.9625	1.68422	0.842109	+	0.539308i	$0.181314\pi$
$564$	0	0
$565$	38.1786	1.60618
$566$	14.9323	0.627653
$567$	0	0
$568$	−10.5859	−0.444174
$569$	−9.87143	−0.413832	−0.206916	−	0.978359i	$-0.566343\pi$
$569$	−9.87143	−0.413832	−0.206916	+	0.978359i	$0.566343\pi$
$570$	0	0
$571$	13.4061	0.561028	0.280514	−	0.959850i	$-0.409495\pi$
$571$	13.4061	0.561028	0.280514	+	0.959850i	$0.409495\pi$
$572$	30.4440	1.27293
$573$	0	0
$574$	2.36381	0.0986635
$575$	−4.59453	−0.191605
$576$	0	0
$577$	−20.1236	−0.837754	−0.418877	−	0.908043i	$-0.637576\pi$
$577$	−20.1236	−0.837754	−0.418877	+	0.908043i	$0.637576\pi$
$578$	−15.2773	−0.635453
$579$	0	0
$580$	6.14206	0.255035
$581$	27.2651	1.13115
$582$	0	0
$583$	62.6624	2.59521
$584$	−13.4736	−0.557543
$585$	0	0
$586$	1.52528	0.0630089
$587$	7.68686	0.317271	0.158635	−	0.987337i	$-0.449291\pi$
$587$	7.68686	0.317271	0.158635	+	0.987337i	$0.449291\pi$
$588$	0	0
$589$	−7.85118	−0.323502
$590$	29.3957	1.21020
$591$	0	0
$592$	8.20682	0.337298
$593$	30.5352	1.25393	0.626965	−	0.779047i	$-0.284297\pi$
$593$	30.5352	1.25393	0.626965	+	0.779047i	$0.284297\pi$
$594$	0	0
$595$	38.7957	1.59047
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−32.5299	−1.33025
$599$	8.61987	0.352198	0.176099	−	0.984372i	$-0.443652\pi$
$599$	8.61987	0.352198	0.176099	+	0.984372i	$0.443652\pi$
$600$	0	0
$601$	−38.6565	−1.57683	−0.788415	−	0.615143i	$-0.789098\pi$
$601$	−38.6565	−1.57683	−0.788415	+	0.615143i	$0.789098\pi$
$602$	−9.89789	−0.403408
$603$	0	0
$604$	11.5048	0.468124
$605$	51.8626	2.10852
$606$	0	0
$607$	−3.78647	−0.153688	−0.0768441	−	0.997043i	$-0.524484\pi$
$607$	−3.78647	−0.153688	−0.0768441	+	0.997043i	$0.524484\pi$
$608$	1.71918	0.0697218
$609$	0	0
$610$	11.3612	0.460001
$611$	−17.5442	−0.709763
$612$	0	0
$613$	−25.4326	−1.02721	−0.513606	−	0.858026i	$-0.671691\pi$
$613$	−25.4326	−1.02721	−0.513606	+	0.858026i	$0.671691\pi$
$614$	3.02894	0.122238
$615$	0	0
$616$	16.2613	0.655186
$617$	48.0612	1.93487	0.967436	−	0.253118i	$-0.0814560\pi$
$617$	48.0612	1.93487	0.967436	+	0.253118i	$0.0814560\pi$
$618$	0	0
$619$	27.8506	1.11941	0.559705	−	0.828692i	$-0.310914\pi$
$619$	27.8506	1.11941	0.559705	+	0.828692i	$0.310914\pi$
$620$	10.9536	0.439906
$621$	0	0
$622$	−11.2549	−0.451280
$623$	13.1814	0.528100
$624$	0	0
$625$	−28.1974	−1.12790
$626$	−29.8744	−1.19402
$627$	0	0
$628$	21.1964	0.845830
$629$	46.6255	1.85908
$630$	0	0
$631$	7.22753	0.287723	0.143862	−	0.989598i	$-0.454048\pi$
$631$	7.22753	0.287723	0.143862	+	0.989598i	$0.454048\pi$
$632$	−13.6549	−0.543164
$633$	0	0
$634$	−25.6497	−1.01868
$635$	12.6589	0.502353
$636$	0	0
$637$	−5.89329	−0.233501
$638$	14.6263	0.579061
$639$	0	0
$640$	−2.39851	−0.0948093
$641$	−15.6213	−0.617004	−0.308502	−	0.951224i	$-0.599828\pi$
$641$	−15.6213	−0.617004	−0.308502	+	0.951224i	$0.599828\pi$
$642$	0	0
$643$	−11.4170	−0.450243	−0.225122	−	0.974331i	$-0.572278\pi$
$643$	−11.4170	−0.450243	−0.225122	+	0.974331i	$0.572278\pi$
$644$	−17.3755	−0.684691
$645$	0	0
$646$	9.76718	0.384284
$647$	−40.5834	−1.59550	−0.797750	−	0.602989i	$-0.793976\pi$
$647$	−40.5834	−1.59550	−0.797750	+	0.602989i	$0.793976\pi$
$648$	0	0
$649$	70.0010	2.74778
$650$	4.01271	0.157392
$651$	0	0
$652$	8.69588	0.340557
$653$	−40.9213	−1.60137	−0.800686	−	0.599084i	$-0.795532\pi$
$653$	−40.9213	−1.60137	−0.800686	+	0.599084i	$0.795532\pi$
$654$	0	0
$655$	−34.9842	−1.36695
$656$	−0.830268	−0.0324165
$657$	0	0
$658$	−9.37104	−0.365321
$659$	−40.3521	−1.57189	−0.785947	−	0.618294i	$-0.787824\pi$
$659$	−40.3521	−1.57189	−0.785947	+	0.618294i	$0.787824\pi$
$660$	0	0
$661$	−35.9942	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$661$	−35.9942	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$662$	14.9538	0.581197
$663$	0	0
$664$	−9.57663	−0.371645
$665$	−11.7396	−0.455244
$666$	0	0
$667$	−15.6285	−0.605137
$668$	−14.1225	−0.546416
$669$	0	0
$670$	−2.82207	−0.109026
$671$	27.0548	1.04444
$672$	0	0
$673$	−51.5161	−1.98580	−0.992899	−	0.118960i	$-0.962044\pi$
$673$	−51.5161	−1.98580	−0.992899	+	0.118960i	$0.962044\pi$
$674$	−22.2886	−0.858523
$675$	0	0
$676$	15.4106	0.592714
$677$	37.2826	1.43288	0.716442	−	0.697646i	$-0.245769\pi$
$677$	37.2826	1.43288	0.716442	+	0.697646i	$0.245769\pi$
$678$	0	0
$679$	25.3199	0.971689
$680$	−13.6267	−0.522559
$681$	0	0
$682$	26.0841	0.998812
$683$	33.0856	1.26598	0.632992	−	0.774158i	$-0.281826\pi$
$683$	33.0856	1.26598	0.632992	+	0.774158i	$0.281826\pi$
$684$	0	0
$685$	31.6617	1.20973
$686$	16.7815	0.640719
$687$	0	0
$688$	3.47655	0.132542
$689$	58.4771	2.22780
$690$	0	0
$691$	−28.0278	−1.06623	−0.533114	−	0.846044i	$-0.678978\pi$
$691$	−28.0278	−1.06623	−0.533114	+	0.846044i	$0.678978\pi$
$692$	5.32450	0.202407
$693$	0	0
$694$	3.31106	0.125686
$695$	39.0323	1.48058
$696$	0	0
$697$	−4.71701	−0.178670
$698$	−19.7113	−0.746085
$699$	0	0
$700$	2.14335	0.0810108
$701$	−10.1345	−0.382773	−0.191386	−	0.981515i	$-0.561298\pi$
$701$	−10.1345	−0.382773	−0.191386	+	0.981515i	$0.561298\pi$
$702$	0	0
$703$	−14.1090	−0.532129
$704$	−5.71165	−0.215266
$705$	0	0
$706$	19.2372	0.724001
$707$	54.7476	2.05899
$708$	0	0
$709$	−3.38964	−0.127300	−0.0636502	−	0.997972i	$-0.520274\pi$
$709$	−3.38964	−0.127300	−0.0636502	+	0.997972i	$0.520274\pi$
$710$	−25.3903	−0.952881
$711$	0	0
$712$	−4.62984	−0.173511
$713$	−27.8713	−1.04379
$714$	0	0
$715$	73.0200	2.73079
$716$	−0.0832261	−0.00311031
$717$	0	0
$718$	4.85892	0.181333
$719$	−8.90668	−0.332163	−0.166081	−	0.986112i	$-0.553111\pi$
$719$	−8.90668	−0.332163	−0.166081	+	0.986112i	$0.553111\pi$
$720$	0	0
$721$	20.1428	0.750158
$722$	16.0444	0.597112
$723$	0	0
$724$	10.0279	0.372683
$725$	1.92784	0.0715983
$726$	0	0
$727$	−7.71690	−0.286204	−0.143102	−	0.989708i	$-0.545708\pi$
$727$	−7.71690	−0.286204	−0.143102	+	0.989708i	$0.545708\pi$
$728$	15.1752	0.562430
$729$	0	0
$730$	−32.3166	−1.19609
$731$	19.7514	0.730531
$732$	0	0
$733$	−3.70541	−0.136862	−0.0684312	−	0.997656i	$-0.521799\pi$
$733$	−3.70541	−0.136862	−0.0684312	+	0.997656i	$0.521799\pi$
$734$	1.25818	0.0464401
$735$	0	0
$736$	6.10300	0.224960
$737$	−6.72029	−0.247545
$738$	0	0
$739$	−4.92413	−0.181137	−0.0905685	−	0.995890i	$-0.528868\pi$
$739$	−4.92413	−0.181137	−0.0905685	+	0.995890i	$0.528868\pi$
$740$	19.6841	0.723602
$741$	0	0
$742$	31.2349	1.14667
$743$	38.5383	1.41383	0.706917	−	0.707297i	$-0.250086\pi$
$743$	38.5383	1.41383	0.706917	+	0.707297i	$0.250086\pi$
$744$	0	0
$745$	−2.39851	−0.0878745
$746$	21.6434	0.792420
$747$	0	0
$748$	−32.4496	−1.18648
$749$	−11.9689	−0.437336
$750$	0	0
$751$	−31.1218	−1.13565	−0.567825	−	0.823150i	$-0.692215\pi$
$751$	−31.1218	−1.13565	−0.567825	+	0.823150i	$0.692215\pi$
$752$	3.29150	0.120029
$753$	0	0
$754$	13.6494	0.497081
$755$	27.5943	1.00426
$756$	0	0
$757$	−23.4397	−0.851930	−0.425965	−	0.904740i	$-0.640065\pi$
$757$	−23.4397	−0.851930	−0.425965	+	0.904740i	$0.640065\pi$
$758$	−24.9958	−0.907889
$759$	0	0
$760$	4.12345	0.149573
$761$	−46.3043	−1.67853	−0.839265	−	0.543722i	$-0.817014\pi$
$761$	−46.3043	−1.67853	−0.839265	+	0.543722i	$0.817014\pi$
$762$	0	0
$763$	−14.5437	−0.526518
$764$	13.2145	0.478083
$765$	0	0
$766$	24.0367	0.868481
$767$	65.3256	2.35877
$768$	0	0
$769$	33.2342	1.19845	0.599227	−	0.800579i	$-0.295475\pi$
$769$	33.2342	1.19845	0.599227	+	0.800579i	$0.295475\pi$
$770$	39.0028	1.40556
$771$	0	0
$772$	10.5122	0.378341
$773$	33.7926	1.21544	0.607718	−	0.794153i	$-0.292085\pi$
$773$	33.7926	1.21544	0.607718	+	0.794153i	$0.292085\pi$
$774$	0	0
$775$	3.43805	0.123499
$776$	−8.89340	−0.319255
$777$	0	0
$778$	7.31780	0.262356
$779$	1.42738	0.0511411
$780$	0	0
$781$	−60.4628	−2.16353
$782$	34.6731	1.23991
$783$	0	0
$784$	1.10565	0.0394875
$785$	50.8398	1.81455
$786$	0	0
$787$	−22.3438	−0.796470	−0.398235	−	0.917283i	$-0.630377\pi$
$787$	−22.3438	−0.796470	−0.398235	+	0.917283i	$0.630377\pi$
$788$	−18.6356	−0.663864
$789$	0	0
$790$	−32.7514	−1.16524
$791$	45.3182	1.61133
$792$	0	0
$793$	25.2478	0.896574
$794$	22.6229	0.802856
$795$	0	0
$796$	−21.2346	−0.752640
$797$	37.1900	1.31734	0.658670	−	0.752432i	$-0.271120\pi$
$797$	37.1900	1.31734	0.658670	+	0.752432i	$0.271120\pi$
$798$	0	0
$799$	18.7001	0.661560
$800$	−0.752832	−0.0266166
$801$	0	0
$802$	38.9060	1.37382
$803$	−76.9566	−2.71574
$804$	0	0
$805$	−41.6752	−1.46886
$806$	24.3419	0.857407
$807$	0	0
$808$	−19.2296	−0.676496
$809$	34.5983	1.21641	0.608207	−	0.793779i	$-0.291889\pi$
$809$	34.5983	1.21641	0.608207	+	0.793779i	$0.291889\pi$
$810$	0	0
$811$	−2.16592	−0.0760558	−0.0380279	−	0.999277i	$-0.512108\pi$
$811$	−2.16592	−0.0760558	−0.0380279	+	0.999277i	$0.512108\pi$
$812$	7.29067	0.255852
$813$	0	0
$814$	46.8744	1.64295
$815$	20.8571	0.730594
$816$	0	0
$817$	−5.97680	−0.209102
$818$	4.29130	0.150042
$819$	0	0
$820$	−1.99140	−0.0695428
$821$	−12.2097	−0.426122	−0.213061	−	0.977039i	$-0.568343\pi$
$821$	−12.2097	−0.426122	−0.213061	+	0.977039i	$0.568343\pi$
$822$	0	0
$823$	−55.3729	−1.93018	−0.965088	−	0.261925i	$-0.915643\pi$
$823$	−55.3729	−1.93018	−0.965088	+	0.261925i	$0.915643\pi$
$824$	−7.07500	−0.246469
$825$	0	0
$826$	34.8929	1.21408
$827$	−47.1721	−1.64033	−0.820167	−	0.572124i	$-0.806120\pi$
$827$	−47.1721	−1.64033	−0.820167	+	0.572124i	$0.806120\pi$
$828$	0	0
$829$	42.3136	1.46961	0.734806	−	0.678277i	$-0.237273\pi$
$829$	42.3136	1.46961	0.734806	+	0.678277i	$0.237273\pi$
$830$	−22.9696	−0.797287
$831$	0	0
$832$	−5.33016	−0.184790
$833$	6.28155	0.217643
$834$	0	0
$835$	−33.8729	−1.17222
$836$	9.81932	0.339608
$837$	0	0
$838$	28.9077	0.998602
$839$	−12.0759	−0.416906	−0.208453	−	0.978032i	$-0.566843\pi$
$839$	−12.0759	−0.416906	−0.208453	+	0.978032i	$0.566843\pi$
$840$	0	0
$841$	−22.4424	−0.773875
$842$	−9.14857	−0.315281
$843$	0	0
$844$	−26.0464	−0.896555
$845$	36.9623	1.27154
$846$	0	0
$847$	61.5613	2.11527
$848$	−10.9710	−0.376746
$849$	0	0
$850$	−4.27708	−0.146702
$851$	−50.0862	−1.71693
$852$	0	0
$853$	20.1310	0.689271	0.344635	−	0.938737i	$-0.388003\pi$
$853$	20.1310	0.689271	0.344635	+	0.938737i	$0.388003\pi$
$854$	13.4858	0.461475
$855$	0	0
$856$	4.20399	0.143690
$857$	9.00187	0.307498	0.153749	−	0.988110i	$-0.450865\pi$
$857$	9.00187	0.307498	0.153749	+	0.988110i	$0.450865\pi$
$858$	0	0
$859$	−0.911692	−0.0311065	−0.0155533	−	0.999879i	$-0.504951\pi$
$859$	−0.911692	−0.0311065	−0.0155533	+	0.999879i	$0.504951\pi$
$860$	8.33853	0.284342
$861$	0	0
$862$	−3.96808	−0.135153
$863$	31.7118	1.07948	0.539741	−	0.841831i	$-0.318522\pi$
$863$	31.7118	1.07948	0.539741	+	0.841831i	$0.318522\pi$
$864$	0	0
$865$	12.7708	0.434221
$866$	7.30067	0.248087
$867$	0	0
$868$	13.0019	0.441315
$869$	−77.9921	−2.64570
$870$	0	0
$871$	−6.27143	−0.212499
$872$	5.10836	0.172991
$873$	0	0
$874$	−10.4921	−0.354901
$875$	−29.0024	−0.980461
$876$	0	0
$877$	39.8191	1.34459	0.672297	−	0.740281i	$-0.265308\pi$
$877$	39.8191	1.34459	0.672297	+	0.740281i	$0.265308\pi$
$878$	−16.0443	−0.541467
$879$	0	0
$880$	−13.6994	−0.461807
$881$	40.7274	1.37214	0.686070	−	0.727535i	$-0.259334\pi$
$881$	40.7274	1.37214	0.686070	+	0.727535i	$0.259334\pi$
$882$	0	0
$883$	−37.7847	−1.27156	−0.635779	−	0.771871i	$-0.719321\pi$
$883$	−37.7847	−1.27156	−0.635779	+	0.771871i	$0.719321\pi$
$884$	−30.2823	−1.01850
$885$	0	0
$886$	1.54926	0.0520484
$887$	−22.5044	−0.755625	−0.377813	−	0.925882i	$-0.623324\pi$
$887$	−22.5044	−0.755625	−0.377813	+	0.925882i	$0.623324\pi$
$888$	0	0
$889$	15.0262	0.503963
$890$	−11.1047	−0.372230
$891$	0	0
$892$	5.34883	0.179092
$893$	−5.65867	−0.189360
$894$	0	0
$895$	−0.199618	−0.00667251
$896$	−2.84704	−0.0951130
$897$	0	0
$898$	−1.09546	−0.0365559
$899$	11.6947	0.390039
$900$	0	0
$901$	−62.3297	−2.07650
$902$	−4.74220	−0.157898
$903$	0	0
$904$	−15.9176	−0.529413
$905$	24.0519	0.799513
$906$	0	0
$907$	−6.35501	−0.211014	−0.105507	−	0.994419i	$-0.533647\pi$
$907$	−6.35501	−0.211014	−0.105507	+	0.994419i	$0.533647\pi$
$908$	12.6760	0.420669
$909$	0	0
$910$	36.3978	1.20657
$911$	40.4910	1.34153	0.670763	−	0.741672i	$-0.265967\pi$
$911$	40.4910	1.34153	0.670763	+	0.741672i	$0.265967\pi$
$912$	0	0
$913$	−54.6983	−1.81025
$914$	−6.70058	−0.221636
$915$	0	0
$916$	15.5735	0.514563
$917$	−41.5264	−1.37132
$918$	0	0
$919$	50.4405	1.66388	0.831940	−	0.554866i	$-0.187231\pi$
$919$	50.4405	1.66388	0.831940	+	0.554866i	$0.187231\pi$
$920$	14.6381	0.482603
$921$	0	0
$922$	32.2767	1.06298
$923$	−56.4244	−1.85723
$924$	0	0
$925$	6.17835	0.203143
$926$	−21.3006	−0.699980
$927$	0	0
$928$	−2.56079	−0.0840619
$929$	26.1812	0.858978	0.429489	−	0.903072i	$-0.358694\pi$
$929$	26.1812	0.858978	0.429489	+	0.903072i	$0.358694\pi$
$930$	0	0
$931$	−1.90081	−0.0622964
$932$	−17.3342	−0.567799
$933$	0	0
$934$	−3.88798	−0.127219
$935$	−77.8307	−2.54534
$936$	0	0
$937$	−40.6778	−1.32889	−0.664443	−	0.747339i	$-0.731331\pi$
$937$	−40.6778	−1.32889	−0.664443	+	0.747339i	$0.731331\pi$
$938$	−3.34981	−0.109375
$939$	0	0
$940$	7.89469	0.257496
$941$	15.4319	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$941$	15.4319	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$942$	0	0
$943$	5.06713	0.165008
$944$	−12.2558	−0.398894
$945$	0	0
$946$	19.8568	0.645601
$947$	30.4059	0.988059	0.494029	−	0.869445i	$-0.335523\pi$
$947$	30.4059	0.988059	0.494029	+	0.869445i	$0.335523\pi$
$948$	0	0
$949$	−71.8166	−2.33126
$950$	1.29425	0.0419910
$951$	0	0
$952$	−16.1749	−0.524233
$953$	32.0933	1.03960	0.519802	−	0.854287i	$-0.326006\pi$
$953$	32.0933	1.03960	0.519802	+	0.854287i	$0.326006\pi$
$954$	0	0
$955$	31.6950	1.02563
$956$	11.9581	0.386753
$957$	0	0
$958$	4.05024	0.130857
$959$	37.5827	1.21361
$960$	0	0
$961$	−10.1441	−0.327229
$962$	43.7436	1.41035
$963$	0	0
$964$	8.21050	0.264442
$965$	25.2135	0.811651
$966$	0	0
$967$	−36.3349	−1.16845	−0.584225	−	0.811592i	$-0.698602\pi$
$967$	−36.3349	−1.16845	−0.584225	+	0.811592i	$0.698602\pi$
$968$	−21.6229	−0.694986
$969$	0	0
$970$	−21.3309	−0.684893
$971$	27.8833	0.894818	0.447409	−	0.894329i	$-0.352347\pi$
$971$	27.8833	0.894818	0.447409	+	0.894329i	$0.352347\pi$
$972$	0	0
$973$	46.3316	1.48532
$974$	−5.04846	−0.161763
$975$	0	0
$976$	−4.73678	−0.151621
$977$	38.3160	1.22584	0.612919	−	0.790145i	$-0.289995\pi$
$977$	38.3160	1.22584	0.612919	+	0.790145i	$0.289995\pi$
$978$	0	0
$979$	−26.4440	−0.845154
$980$	2.65191	0.0847121
$981$	0	0
$982$	22.2298	0.709383
$983$	1.95610	0.0623900	0.0311950	−	0.999513i	$-0.490069\pi$
$983$	1.95610	0.0623900	0.0311950	+	0.999513i	$0.490069\pi$
$984$	0	0
$985$	−44.6975	−1.42418
$986$	−14.5486	−0.463323
$987$	0	0
$988$	9.16347	0.291529
$989$	−21.2174	−0.674674
$990$	0	0
$991$	50.0986	1.59143	0.795717	−	0.605668i	$-0.207094\pi$
$991$	50.0986	1.59143	0.795717	+	0.605668i	$0.207094\pi$
$992$	−4.56683	−0.144997
$993$	0	0
$994$	−30.1385	−0.955934
$995$	−50.9313	−1.61463
$996$	0	0
$997$	−25.2996	−0.801248	−0.400624	−	0.916243i	$-0.631207\pi$
$997$	−25.2996	−0.801248	−0.400624	+	0.916243i	$0.631207\pi$
$998$	−29.4416	−0.931957
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.q.1.11	✓	14
3.2	odd	2		8046.2.a.r.1.4	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.11	✓	14	1.1	even	1	trivial
8046.2.a.r.1.4	yes	14	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.39851 q^{5} +2.84704 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.39851 q^{5} +2.84704 q^{7} -1.00000 q^{8} -2.39851 q^{10} -5.71165 q^{11} -5.33016 q^{13} -2.84704 q^{14} +1.00000 q^{16} +5.68131 q^{17} -1.71918 q^{19} +2.39851 q^{20} +5.71165 q^{22} -6.10300 q^{23} +0.752832 q^{25} +5.33016 q^{26} +2.84704 q^{28} +2.56079 q^{29} +4.56683 q^{31} -1.00000 q^{32} -5.68131 q^{34} +6.82865 q^{35} +8.20682 q^{37} +1.71918 q^{38} -2.39851 q^{40} -0.830268 q^{41} +3.47655 q^{43} -5.71165 q^{44} +6.10300 q^{46} +3.29150 q^{47} +1.10565 q^{49} -0.752832 q^{50} -5.33016 q^{52} -10.9710 q^{53} -13.6994 q^{55} -2.84704 q^{56} -2.56079 q^{58} -12.2558 q^{59} -4.73678 q^{61} -4.56683 q^{62} +1.00000 q^{64} -12.7844 q^{65} +1.17659 q^{67} +5.68131 q^{68} -6.82865 q^{70} +10.5859 q^{71} +13.4736 q^{73} -8.20682 q^{74} -1.71918 q^{76} -16.2613 q^{77} +13.6549 q^{79} +2.39851 q^{80} +0.830268 q^{82} +9.57663 q^{83} +13.6267 q^{85} -3.47655 q^{86} +5.71165 q^{88} +4.62984 q^{89} -15.1752 q^{91} -6.10300 q^{92} -3.29150 q^{94} -4.12345 q^{95} +8.89340 q^{97} -1.10565 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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